1,088 research outputs found
Spectrum of mixed bi-uniform hypergraphs
A mixed hypergraph is a triple , where is
a set of vertices, and are sets of hyperedges. A
vertex-coloring of is proper if -edges are not totally multicolored and
-edges are not monochromatic. The feasible set of is the set of
all integers, , such that has a proper coloring with colors.
Bujt\'as and Tuza [Graphs and Combinatorics 24 (2008), 1--12] gave a
characterization of feasible sets for mixed hypergraphs with all - and
-edges of the same size , .
In this note, we give a short proof of a complete characterization of all
possible feasible sets for mixed hypergraphs with all -edges of size
and all -edges of size , where . Moreover, we show that
for every sequence , , of natural numbers there
exists such a hypergraph with exactly proper colorings using colors,
, and no proper coloring with more than colors. Choosing
this answers a question of Bujt\'as and Tuza, and generalizes
their result with a shorter proof.Comment: 9 pages, 5 figure
The chromatic spectrum of 3-uniform bi-hypergraphs
Let be a finite set of positive integers with
and . For any positive integers , we
construct a family of 3-uniform bi-hypergraphs with the feasible set
and , where each is the th
component of the chromatic spectrum of . As a result, we solve one
open problem for 3-uniform bi-hypergraphs proposed by Bujt\'{a}s and Tuza in
2008. Moreover, we find a family of sub-hypergraphs with the same feasible set
and the same chromatic spectrum as it's own. In particular, we obtain a small
upper bound on the minimum number of vertices in 3-uniform bi-hypergraphs with
any given feasible set
-WORM colorings of graphs: Lower chromatic number and gaps in the chromatic spectrum
A -WORM coloring of a graph is an assignment of colors to the
vertices in such a way that the vertices of each -subgraph of get
precisely two colors. We study graphs which admit at least one such
coloring. We disprove a conjecture of Goddard et al. [Congr. Numer., 219 (2014)
161--173] who asked whether every such graph has a -WORM coloring with two
colors. In fact for every integer there exists a -WORM colorable
graph in which the minimum number of colors is exactly . There also exist
-WORM colorable graphs which have a -WORM coloring with two colors
and also with colors but no coloring with any of colors. We
also prove that it is NP-hard to determine the minimum number of colors and
NP-complete to decide -colorability for every (and remains
intractable even for graphs of maximum degree 9 if ). On the other hand,
we prove positive results for -degenerate graphs with small , also
including planar graphs. Moreover we point out a fundamental connection with
the theory of the colorings of mixed hypergraphs. We list many open problems at
the end.Comment: 18 page
Generalisation : graphs and colourings
The interaction between practice and theory in mathematics is a central theme. Many mathematical structures and theories result from the formalisation of a real problem. Graph Theory is rich with such examples. The graph structure itself was formalised by Leonard Euler in the quest to solve the problem of the Bridges of Königsberg. Once a structure is formalised, and results are proven, the mathematician seeks to generalise. This can be considered as one of the main praxis in mathematics. The idea of generalisation will be illustrated through graph colouring. This idea also results from a classic problem, in which it was well known by topographers that four colours suffice to colour any map such that no countries sharing a border receive the same colour. The proof of this theorem eluded mathematicians for centuries and was proven in 1976. Generalisation of graphs to hypergraphs, and variations on the colouring theme will be discussed, as well as applications in other disciplines.peer-reviewe
- …